ClippedScion

Code accompanying the paper Generalized Gradient Norm Clipping & Non-Euclidean $(L_0,L_1)$-Smoothness.

This paper is a following work of Training Deep Learning Models with Norm-Constrained LMOs and is based on the Scion codebase.

Repository structure

• clippedscion.py: Contains the UnconstrainedClippedScion and ClippedScion reference implementation along with various norm choices.
  • Algorithm 3 corresponds to UnconstrainedClippedScion.
  • Algorithm 4 (Variant 2) corresponds to ClippedScion. For simplicity, we control $\min{\rho, \sum_{l=1}^D \braket{d^k_l,v^k_l}}$ in practice.
• examples/: Example usage containing airbench, nanoGPT, and DeiT experiments with and without weight sharing.

Notes

The ClippedScion optimizer comes with a couple of hyperparameters:

• momentum: The parameter is 1-usual_momentum of e.g. the PyTorch implementation of SGD with momentum. A good default is 0.1. Higher values seem to work better (e.g. 0.5) for short training runs with low noise as also supported by theory.
• scale: Controls the per-layer constraint radius factor. The layerwise radius can be tuned on a small proxy model similarly to the input and output scaling factor of µP.
• lr: The learning rate can similarly be tuned on a small proxy model (corresponds to γ in the paper).
• unconstrained: When set to False the constrained variant of the ClippedScion is used, which guarantees the iterates to stay bounded.
• rho: Clipping threshold controls $\sum_{l=1}^D \braket{d^k_l,v^k_l}$ in Algorithm 3 & 4.

Architectural changes:

• Scale activation functions (ReLU, GELU) by √2 to maintain the input variance.

Examples

For runnable examples see examples/. Below are some pseudocode configurations for different architectures and domains (see Appendix C for exact parameter choices):

• nanoGPT with weight sharing (see examples/modded-nanogpt):

  radius = 50.0
  threshold = 600
  optim_groups = [{
      'params': model.transformer.h.parameters(),
      'norm': 'Spectral',
      'norm_kwargs': {},
      'scale': radius,
  }, {
      'params': model.lm_head.parameters(),
      'norm': 'Sign',
      'norm_kwargs': {},
      'scale': radius*60.0,
  }]
  optimizer = UnconstrainedClippedScion(optim_groups, lr=2**-12, momentum=0.1, rho=600)

• CNN (see examples/airbench for further details):

  radius = 8.0
  threshold = 1600
  optim_groups = [{
      'params': remaining_parameters,
      'norm': 'Auto', # Picks layerwise norm based on the parameter shape
      'norm_kwargs': {},
      'scale': radius,
  }, {
      'params': output_layer,
      'norm': 'Sign',
      'norm_kwargs': {'normalized': True},
      'scale': radius*16,
  }]
  optimizer = UnconstrainedClippedScion(optim_groups, lr=2**-4, momentum=0.5, rho=1600)

• DeiT

  radius = 25
  threshold = 8000
  optim_groups = [{
      'params': other_params,
      'norm': 'Auto',
      'norm_kwargs': {},
      'scale': radius,
  },{
      'params': head_weights,
      'norm': 'Sign',
      'norm_kwargs': {},
      'scale': radius*20,
  },{
      'params': [pos_embed_param, cls_token_param],
      'norm': 'BiasRMS',
      'norm_kwargs': {},
      'scale': radius,
  }]
  optimizer = UnconstrainedClippedScion(optim_groups, lr=8e-5, momentum=0.1, rho=8000)

Citation

If you find this work useful, please cite it as follows:

@article{pethick2025generalized,
  title={Generalized Gradient Norm Clipping \& Non-Euclidean $(L\_0, L\_1) $-Smoothness},
  author={Pethick, Thomas and Xie, Wanyun and Erdogan, Mete and Antonakopoulos, Kimon and Silveti-Falls, Tony and Cevher, Volkan},
  journal={arXiv preprint arXiv:2506.01913},
  year={2025}
}